Numerical Analysis

1. Numerical Analysis EE, NCKU Tien-Hao Chang (Darby Chang)

2. In the previous slide • Why numerical methods? • differences between human and computer • a very simple numerical method • What is algorithm? • definition and components • three problems and three algorithms • Convergence • compare rate of convergence

3. In this slide • Error (motivation) • Floating point number system • difference to real number system • problem of roundoff • Introduced/propagated error • Focus on numerical methods • three bugs

4. Let’s start from error • Numerical methods are generally designed to determine approximation solutions • 3 categories of error types • modeling: made when you decide the algorithm • discretization/truncation: conversion from continuous to discrete and/or truncation of an infinite series • roundoff/data: not due to the formulation of a numerical method, caused by the data representation (in computer)

5. Can be analyzed • Numerical methods are generally designed to determine approximation solutions • 3 categories of error types • modeling: made when you decide the algorithm • discretization/truncation: conversion from continuous to discrete and/or truncation of an infinite series • roundoff/data: not due to the formulation of a numerical method, caused by the data representation (in computer)

6. Should be prevented • Numerical methods are generally designed to determine approximation solutions • 3 categories of error types • modeling: made when you decide the algorithm • discretization/truncation: conversion from continuous to discrete and/or truncation of an infinite series • roundoff/data: not due to the formulation of a numerical method, caused by the data representation (in computer)

7. 1.3 Floating Point Number Systems

9. Restriction of must not be zero (except when the number being represented is 0)

10. Floating point vs. real number • Discrete vs. continuous • continuous means that between any two numbers, there are infinitely many other numbers • Finite vs. infinite • number of element and range of values • a floating point number system contains its smallest/largest element • underflow/overflow

12. Floating point vs. real number • Nonuniform vs. uniform • real numbers are uniformly distributed • in a floating point number system, the elements **** *** **** are more closely spaced • think about the difference between two adjacent elements while the exponent changes hint

13. Floating point vs. real number • Nonuniform vs. uniform • real numbers are uniformly distributed • in a floating point number system, the elements **** *** **** are more closely spaced • think about the difference between two adjacent elements while the exponent changes

14. Floating point vs. real number • Nonuniform vs. uniform • real numbers are uniformly distributed • in a floating point number system, the elements near the zero are more closely spaced • think about the difference between two adjacent elements while the exponent changes

15. Floating point system is discrete, finite and nonuniform

16. Roundoff error • When the number is outside the system • Select an element to represent the number • chop • round • A number to its floating point equivalent • y→fl(y)

19. Roundoff error • When the number is outside the system • Select an element to represent the number • chop • round • A number to its floating point equivalent

20. Formal definition

21. An example

22. In general case (chopped)

23. In general case (chopped)

24. Machine precision/epsilon • The error bound is independent of the number, y • It depends on • base () • the number of digits (k) • The bound is a function of the hardware implementation • Cause of roundoff error

25. Formal definition

26. Another term about precision

28. So far, we talked about floating point number systems in abstract

29. Then, what systems are we likely to encounter in practice?

30. Real floating point system • 1970s • begun to develop a standard binary floating point numbers to eliminate inconsistencies • 1985 • IEEE • Binary Floating Point Arithmetic Standard 754 • The IEEE Standard • F(2,24,-125,128), single precision • F(2,53,-1021,1024), double precision

31. IEEE standard single precision

32. 1.4 Floating Point Arithmetic

33. Motivation • Floating point arithmetic stands for the mathematics on the computer, but why should we know that? • The IEEE Standard • seems pretty accurate • However,

34. Numerical methods perform a sequence of calculations on computer, where each operation introduces some roundoff error

35. http://www.radgraphics.net/images/main/atomic%20explosion%20-%204.jpghttp://www.radgraphics.net/images/main/atomic%20explosion%20-%204.jpg when they are accumulated

36. Typical arithmetic • Three steps • operand its floating point equivalent • the exact arithmetic • result its floating point equivalent

38. Not associative • We should perform the arithmetic in ********* order to obtain the most accurate result question

39. All intermediate results have been rounded

41. Not associative • We should perform the arithmetic in ********* order to obtain the most accurate result

42. Not associative • We should perform the arithmetic in Ascending order to obtain the most accurate result

43. In FP arithmetic, always notice the number of significant digits and the least significant bits

44. Not distributive

45. Accumulation of roundoff error

47. Introduced/propagated error

48. Propagated error can be large even if the introduced error is small

49. A notation in the analysis

50. In multiplication